Analysts on harmonic maps over geometric singular spaces via Dirichlet forms

通过狄利克雷形式分析几何奇异空间上的调和映射

基本信息

批准号：
16540201
负责人：
KUWAE Kazuhiro
金额：
$ 2.37万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540201/
关键词：
Dirichlet form harmonic map subharmonic function variational convergence Gromov-Hausdorff convergence Alexandrov space Kato class heat kennel アレキサンドロフ空間

项目摘要

We establish the following result :1) Variational convergence of metric measure spaces:We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya.2) Perturbation of symmetric Markov processes and its related stocha … More stic calculus:We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang.3) Kato class measures over symmetric Markov processes :We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi.4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes:We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato.5) Liouville theorems for harmonic maps to convex spaces over Markov chains:We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm.6) Laplacian comparison theorem on Alexandrov spaces :We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya. Less

我们建立以下结果：1）度量空间的变化收敛：我们侵入了地图的LP连接的定义。拓扑和目标是与LP相关的Gromov-Hausdorff序列。 suburevel。我们提出了一个路径空间的积分代表与一般对称Markov过程的L2-IniTimal Generator L相关的半群，用于对称Markov过程的零能量构成。。（x; m）$表示$ | f |。（x; m）$。连续的添加功能（n，p） - sato.5进行。在保守的营销方面的功能，可以使用安德罗夫教授进行，这项工作是在安德罗夫教授中进行的。下面，我们证明了拓扑成绩